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Digital Image Processing Basics

Module by: Robert Nowak

Summary: The module provides an introduction to the concepts of digital imaging processing through basic equations and examples.

Digital Image Processing

A sampled image gives us our usual 2D array of pixels fmn f m n (Figure 1):

Figure 1: We illustrate a "pixelized" smiley face.
Figure 1 (smiley.png)

We can filter fmn f m n by applying a 2D discrete-space convolution as shown below (where hmn h m n is our PSF):

gmn=hmn*fmn=∑k=-∞∞∑l=-∞∞hm-kn-lfkl g m n h m n f m n k l h m k n l f k l (1)

Example 1: Sampled Image

Figure 2: Illustrate the "pixelized" nature of all digital images.
Figure 2 (sec2_eg1.png)

We also have discrete-space FTS:

Fuv=∑m=-∞∞∑n=-∞∞fmnⅇ-ⅈumⅇ-ⅈvm F u v m n f m n u m v m (2)
where Fuv F u v is analogous to DTFT in 1D.

note:

"Convolution in Time" is the same as "Multiplication in Frequency"
gmn=hmn*fmn g m n h m n f m n (3)
which, as stated above, is the same as:
Guv=HuvFuv G u v H u v F u v (4)

Example 2: Magnitude of FT of Cameraman Image

Figure 3
Figure 3 (cam_mag.png)

To get a better image, we can use the fftshift command in Matlab to center the Fourier Transform. The resulting image is shown in Figure 4:

Figure 4
Figure 4 (cam_mag_center.png)

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